Picards Existence and Uniqueness theorem doesn't prove anything

[Al3105]

As far as I can understand it, Picard's Existence and Uniqueness of ODEs theorem relies on the fact that a the given function f(x,t) in the initial value problem dx/dt = f(x,t) x(t0) = x0 is Lipschitz continuous and bounded on a rectangular region of the plane that it's defined on. And the theorem(mainly through successive approximations) proves that the solution exists and is unique ONLY in that particular region.

I believe that there exists a theorem which proves, that the Picard theorem implies existence and uniqueness for the whole Interval that the given function is defined on. However I can not for the life of me find that theorem.

So how does this work? Am I correct in what I stated above? If I am, then why isn't that theorem as easy to find as Picard's theorem? It would seem that it's just as important.

Could someone explain the theorem and perhaps give or help me find a proof for it?

 

[gtoguy97]

The Picard–Lindelöf theorem (also called the Picard–Lindelöf existence and uniqueness theorem) states that if a given initial value problem in an ordinary differential equation has a continuous right-hand side, i.e., f(x,t) is continuous for all (x,t) in a rectangular region of the x-t plane, then it has a unique solution. This theorem is a generalization of the Picard's Existence and Uniqueness theorem which requires that f(x,t) be Lipschitz continuous on a rectangular region of the plane. The proof of this theorem follows from the fact that if a function is continuous, then it is locally Lipschitz continuous, and thus satisfies the conditions of the Picard's Existence and Uniqueness theorem. As such, the solution to the differential equation exists and is unique in any subregion of the rectangular region of the x-t plane, and hence, by the principle of analytic continuation, also exists and is unique in the entire rectangular region. In other words, Picard–Lindelöf theorem states that if the right-hand side of an initial value problem in an ordinary differential equation is continuous, then the solution exists and is unique on the entire interval on which the differential equation is defined.